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Theory of Complex Functions (Graduate Texts in Mathematics) (v. 122)
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Review
R. Remmert and R.B. Burckel Theory of Complex Functions "Its accessibility makes it very useful for a first graduate course on complex function theory, especially where there is an opportunity for developing an interest on the part of motivated students in the history of the subject. Historical remarks abound throughout the text. Short biographies of Abel, Cauchy, Eisenstein, Euler, Riemann and Weierstrass are given. There is an extensive bibliography of classical works on complex function theory with comments on some of them. In addition, a list of modern complex function theory texts and books on the history of the subject and of mathematics is given. Throughout the book there are numerous interesting quotations. In brief, the book affords splendid opportunities for a rich treatment of the subject."―MATHEMATICAL REVIEWS
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Language Notes
Text: English (translation) Original Language: German
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Product details
Series: Graduate Texts in Mathematics (Book 122)
Hardcover: 458 pages
Publisher: Springer (December 21, 1998)
Language: English
ISBN-10: 0387971955
ISBN-13: 978-0387971957
Product Dimensions:
6.1 x 1.1 x 9.2 inches
Shipping Weight: 1.8 pounds (View shipping rates and policies)
Average Customer Review:
4.3 out of 5 stars
5 customer reviews
Amazon Best Sellers Rank:
#1,278,496 in Books (See Top 100 in Books)
This review applies equally to the sequel "Classical Topics in Complex Function Theory".Out of all the introductory complex analysis texts I've encountered, this is my favorite.For a brief overview according to me: Conway's book covers the right topics but the presentation is very jumbled and reading it is quite tedious. Ahlfor's book is very elegantly written but its explanations lack the detail most beginning students need. The Saff-Snider book is excellent for many computational aspects such as Laurent series, integration using residues, and conformal mapping (even for the graduate level) but doesn't include the deeper topics. A more recent book I'd highly recommend as a supplement is "Visual Complex Functions" by Wegert, which gives a solid introduction to phase portraits (I consider this is an upgrade over the Needham "Visual Complex Analysis" book, which is a fine book but just not as insightful in my opinion as it tries to be).Back to this book: It is very well-organized, and the writing is simultaneously simple and straight-forward while also brimming with insight. The proofs are generally intuitive and natural, and Remmert usually succeeds at drawing attention to the key underlying principles, as a mathematician would see it. Two examples that stand out to me (among many others) are the Identity Theorem and the proof of existence of logarithms--I'm sure other books cover these topics, but they suddenly became a lot clearer in my mind after reading this book. The arrangement of topics is somewhat nonstandard, yet I tend to prefer Remmert's arrangement--the different pieces of the theory are developed very naturally.The biggest strength, however, is the detailed historical commentary after all the main theorems. You will learn about who proved the theorem first, the gist of the original proof, how subsequent mathematicians refined it, and in some cases some feuding between mathematicians. I personally find this incredibly interesting, and pedagogically-speaking it helps you learn the material better by personalizing it. In my opinion this is a practice every math textbook should follow though very few do. The next best I've seen (at least as far as standard textbooks go) is Folland's "Real Analysis", but even in this case the historical notes are relegated to the end of each chapter.As already noted, the main weakness is the lack of exercises. But exercises are not hard to find, for example, by finding old complex analysis qualifying exams from your university. So I don't consider this a signficant strike against it.
This is the english translation of a german textbook. Among its qualities are: very good order of subjects, a good synthesis of classical results and theorems, an open view on current research in the field, glimpses of functions of several complex variables and historical notes.This covers all the basic material needed for a graduate in mathematics preparing him/her to read the sequel: "Classical Topics in Complex Function Theory". Same author (with Grauert), same editor.My only regret is the relatively few exercices compared to other books, but this quite easy to find anyway in the litterature.Most quotations are both in the original language and translated.
This book is a beautiful look at classical complex analysis. At first glance it may look as if it doesn't cover that much, but if you read it carefully you will see that it is filled with useful information. There are many great books on the subject, and I would count this among them. For comparison, Alhfors is wonderful, but it is a bit dry. Remmert's book is not dry at all though, and draws you in as soon as you pick it up.
In a nutshell, it is good indeed as a leisurely reading, at bedtime, as the series description suggests. It should NOT be used as a text, in my opinion.The most important quality of a good graduate level text is usability for a qualifying exam preparation. Thus, everything should be covered in a concise manner, appropriate for late night crumming. This book is not designed for that. And, with all due respect to Goethe, Schiller, and Heine, I could not appreciate an accidental quotation from a classic when my stay in grad school was at stake.(for the record, I passed with flying colors, only thanks to good old references by Conway and Ahlfors, however dry but perfectly readable, especially if you have a prior experience or a helpful professor)Another point to consider: a price can be a factor for a broke grad student, so, Ahlfors will give you more stuff for your buck, especially if you are planning on taking a second semester of this amazing topic.
This is one of the best mathematical books that I ever read!
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